Mechanical stabilization of the Levitron’s realistic model

Abstract

The stability of the magnetic levitation showed by the Levitron was studied by M.V. Berry as a six degrees of freedom Hamiltonian system using an adiabatic approximation. Further, H.R. Dullin found critical spin rate bounds where the levitation persists and R.F. Gans et al. offered numerical results regarding the initial conditions’ manifold where this occurs. In the line of this series of works, first, we extend the equations of motion to include  dissipation for a more realistic model, and then introduce a mechanical forcing to inject energy into the system in order to prevent the Levitron from falling. A systematic study of the flying time as a function of the forcing parameters is carried out which yields detailed bifurcation diagrams showing an Arnold’s tongues structure.  The stability of these solutions were studied with the help of a novel method to compute the maximum Lyapunov exponent called MEGNO. The bifurcation diagrams for MEGNO reproduce the same Arnold’s tongue structure.